Pressurized Water Reactor (PWR)

Models:

•  Involve neutron transport, thermal-hydraulics, chemistry

•  Inherently multi-scale, multi-physics CRUD Measurements: Consist of low resolution images at limited number of locations.

Pressurized Water Reactor (PWR)

3-D Neutron Transport Equations:

Challenges:

•  Linear in the state but function of 7 independent variables:

•  Very large number of inputs or parameters; e.g., 100,000; Parameter selection critical.

•  ORNL Code: Denovo •  Codes can take hours to days to run.

Pressurized Water Reactor (PWR) Thermo-Hydraulic Model: Mass, momentum and energy balance for fluid

Challenges:

•  Nonlinear coupled PDE with nonphysical parameters due to closure relations;

•  CASL code: COBRA (CTF) •  COBRA is a sub-channel code, which cannot be resolved between pins. •  Codes can take minutes to days to run.

Note: Similar equations for gas

Microscopic Cross-Sections

Cross-Sections: Probability that a neutron-nuclei reaction will occur is characterized by nuclear cross-sections.

•  Assume target is sufficiently thin so no shielding. •  Rate of Reactions:

•  Microscopic Cross-Sections: Related to types of reactions.

Flux I

NA

neutrons

cm

2 · s

nuclei

cm2R = �INA

Note: � is microscopic cross-section (cm2)

�f : Fission

�s: Scatter�t: Total

Reference: J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis, John Wiley and Sons, 1976.

Macroscopic Cross-Sections

Macroscopic Cross-Sections: Accounts for shielding

Total Reaction Rate per Unit Area:

dR = �tIdNA

= �tINdx

Strategy: Equate reaction rate to decrease in intensity

�dI(x) = �[I(x+ dx)� I(x)]= �tINdx

) dIdx

= �N�tI(x)

) I(x) = I0e�N�tx

Total Macroscopic Cross-Section: ⌃t ⌘ N�tFrequency: With which reactions occur v⌃t

cm

s

1

cm=

1

s

I0

Neutron Flux in Reactor

Neutron Density: N(r, t)

• N(r, t)dr3: Expected number of neutrons in dr3 about r at time t

Reaction Rate Density: Interaction frequency

dr3

r

• N(r, E, t)dr3dE: Expected number neutrons in dr3 with energies E in dE

Angular Neutron Density:

• n(r, E, ⌦̂, t)dr3dEd⌦̂: Expected number of neutrons in dr3 about rwith energy E about dE, moving in direction ⌦̂

in solid angle ⌦̂ at time t

Angular Neutron Flux: '(r, E, ⌦̂, t) = vn(r, E, ⌦̂, t)

F (r, E, t)dr3 = v⌃N(r, E, t)dEdr3

v⌃ where v is neutron speed

Neutron Transport Equation Conservation Law:

Gain Mechanisms: (1) Neutron sources (fission)

(2) Neutrons entering V

(3) Neutrons of different E0, ⌦̂0 that change to E, ⌦̂ due to scattering collision

Loss Mechanisms: (4) Neutrons leaving V

(5) Neutrons suffering a collision

@

@t

Z

Vn(r, E, ˆ⌦, t)dr3

�= gain in V � loss in V

Neutron Transport Equation Terms:

(1)

Z

VS(r, E, ⌦̂, t)dr3dEd⌦̂ where S is a source term

(5) Rate at which neutrons collide at point r is

) (5) =hR

V vP

t(r, E)n(r, E, ⌦̂, t)dr3idEd⌦̂

j(r, E, ⌦̂, t) · dS = v⌦̂n(r, E, ⌦̂, t) · dSso leakage is

=

Z

Vr · v⌦̂n(r, E, ⌦̂, t)dr3

�dEd⌦̂

=

Z

Vv⌦̂ ·rn(r, E, ⌦̂, t)dr3

�dEd⌦̂

(2), (4) Consider the rate at which neutrons leak out of surface

(4) - (2) =

Z

SdS · v⌦̂n(r, E, ⌦̂, t)

�dEd⌦̂

ft = v⌃t(r, E)n(r, E, ⌦̂, t)

Neutron Transport Equation Terms: Scattering cross-sections

•  Consider first a beam of neutrons of incident intensity I, all of energy E’, hitting a thin target of surface atomic density NA

E0 E

Note: Microscopic differential scattering cross-section is proportionality constant

Rate

cm2= �s(E

0 ! E)INAdE

Microscopic scattering cross-section:

�s(E0) =

Z 1

0dE�s(E

0 ! E)

when neutron scatters from energy E0 to final energy E in range E to E+dE

Macroscopic scattering cross-section:

⌃s(E0 ! E) ⌘ N�s(E0 ! E)

Neutron Transport Equation Terms: Scattering cross-sections

•  Consider how change in direction

⌦̂0 ⌦̂�s(⌦̂

0) =

Z

4⇡d⌦̂�s(⌦̂

0 ! ⌦̂)

(3) Rate at which neutrons scatter from E0, ⌦̂0 to E, ⌦̂ isZ

Vv0⌃s(E

0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t)dr3�dEd⌦̂

To incorporate contributions from any E0, ⌦̂0, we integrate to obtain

(3) =

Z

Vdr3

Z

4⇡d⌦̂0

Z 1

0dE0v0⌃s(E

0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t)�dEd⌦̂

Neutron Transport Equation Conservation Law:

Since this must hold for any control volume, it follows that @n

@t+ v⌦̂ ·rn+ v⌃tn(r, E, ⌦̂, t)

0 =

Z

V

@n

@t+ v⌦̂ ·rn+ v⌃tn(r, E, ⌦̂, t)

Angular Flux: '(r, E, ⌦̂, t) = vn(r, E, !̂, t)

1

v

@'

@t+ ⌦̂ ·r'+ ⌃t(r, E)'(r, E, ⌦̂, t)

�Z 1

0dE0

Z

4⇡d⌦̂0v0⌃s(E

0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t)�S(r, E, ⌦̂, t)�dr3dEd⌦̂

=

Z

4⇡d⌦̂0

Z 1

0dE0⌃s(E

0 ! E, ⌦̂0 ! ⌦̂)'(r, E0, ⌦̂0, t) + S(r, E, ⌦̂, t)

=

Z

4⇡d⌦̂0

Z 1

0dE0v0⌃s(E

0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t) + S(r, E, ⌦̂, t)

Neutron Transport Equation Plane Symmetry: Flux depends only on x

x

⌦̂

✓ = cos�1 µ• ⌦x

= cos ✓

•Z

4⇡d⌦̂0 =

Z ⇡

0d✓0 sin ✓0

Then ˆ

⌦ ·r' = ⌦x

@'

@x

= cos ✓

@'

@x

so 1

v

@'

@t

+ cos ✓

@'

@x

+ ⌃t'(x,E, ✓, t)

=

Z ⇡

0d✓

0 sin ✓0Z 1

0dE

0⌃s(E0 ! E, ✓0 ! ✓)'(x,E0, ✓0, t) + S(x,E, ✓, t)

1-D Neutron Transport Equation: 1

v

@'

@t

+ µ@'

@x

+ ⌃t'(x,E, µ, t)

=

Z 1

�1dµ

0Z 1

0dE

0⌃s(E0 ! E, µ0 ! µ)'(x,E0, µ0, t) + S(x,E, µ, t)